When you move from one state to another, the general rule is that you become a resident of the new state on the day you begin living in your new permanent home (your new domicile) in the new state. So I would use the date in July when you abandoned your MA domicile and made the NY residence your new permanent home. Your income from that date forward is entirely taxable by NY. Your employer should be withholding NY taxes. If they cannot withhold NY taxes, you should begin making quarterly estimated tax payments to NY.
For 2020 you will file a part-year resident tax return in each of the two states. Note that the income you earned by working remotely in NY prior to its becoming your state of residence is taxable by NY, as well as by your then-domicile state of MA. So the income you report on your part-year resident NY tax return must include both your NY-source income from your period of non-residence as well as all your income after becoming an NY resident. (NY-source income is income you earn from work actually performed in NY.) See page 18 of this web reference for NY's instructions on how to allocate income in your situation: https://www.tax.ny.gov/pdf/current_forms/it/it203i.pdf
On your part-year MA return, you'll be able to claim a credit for the tax paid to NY on the portion of your income that is taxable by both states, so you won't be double-taxed. Your income after becoming an NY resident is not taxable by MA unless you physically work inside MA.
**Answers are correct to the best of my ability but do not constitute tax or legal advice.
